Step 4: Convergence to a weak solution.
In view of the uniform estimates (REF ) and (), the Banach–Alaoglu theorem and the Aubin–Lions–Simon lemma (see, e.g., [1]}) imply the existence of functions \({v}\) , \(\phi \) , \(\Psi \) , \(\mu \)  and \(\theta \) 
such that
\(\partial _{t}{v}_k &\rightarrow \partial _{t}{v}&&\quad \text{weakly in $L^\frac{4}{3}(0,T;\mathbf {H}^1_\textnormal {div}(\Omega )^{\prime })$ if $d=3$,}\\&&&\quad \text{weakly in $L^2(0,T;\mathbf {H}^1_\textnormal {div}(\Omega )^{\prime })$ if $d=2$,}\\[1ex]{v}_k &\rightarrow {v}&&\quad \text{weakly-star in $L^\infty (0,T;\mathbf {L}_\textnormal {div}^2(\Omega ))$,}\\&&&\quad \text{weakly in $L^2(0,T;\mathbf {H}^1(\Omega )) \cap L^2(0,T;\mathbf {L}^2(\Gamma ))$,}\\&&&\quad \text{strongly in $L^2(0,T; \mathbf {L}_\textnormal {div}^2(\Omega ))\cap C([0,T];\mathbf {H}^1_\textnormal {div}(\Omega )^{\prime })$,}\\[1ex](\partial _{t}\phi _k,\partial _{t}\Psi _k) &\rightarrow (\partial _{t}\phi ,\partial _{t}\Psi )&&\quad \text{weakly in $L^2(0,T;\mathcal {D}_\alpha ^{\prime })$,}\\[1ex](\phi _k,\Psi _k) &\rightarrow (\phi ,\Psi )&&\quad \text{weakly-star in $L^\infty (0,T;{\mathcal {D}_{\alpha }})$,}\\&&&\quad \text{strongly in $C([0,T];\mathcal {H}^s)$ for all $s\in [0,1)$,}\\[1ex](\mu _k,\Theta _k) &\rightarrow (\mu ,\Theta )&&\quad \text{weakly in $L^4(0,T;\mathcal {L}^2)\cap L^2(0,T;{\mathcal {D}_{\alpha }})$,}\) 
